Weak complicial sets are simplicial sets with extra structure that are closely related to the ω-nerves of weak ω-categories.
The goal of characterizing such nerves, without an a priori definition of “weak $\omega$-category” to start from, is called simplicial weak ω-category theory. It is expected that the (nerves of) weak $\omega$-categories will be weak complicial sets satisfying an extra “saturation” condition ensuring that “every equivalence is thin.” General weak complicial sets can be regarded as “presentations” of weak $\omega$-categories.
Weak complicial sets are a joint generalization of
Let
$\Delta^k[n]$ be the stratified simplicial set whose underlying simplicial set is the $n$-simplex $\Delta[n]$, and whose marked cells are precisely those simplices $[r] \to [n]$ that contain $\{k-1, k, k+1\} \cap [n]$;
$\Lambda^k[n]$ be the stratified simplicial set whose underlying simplicial set is the $k$-horn of $\Delta[n]$, with marked cells those that are marked in $\Delta^k[n]$;
$\Lambda^k[n]'$ be obtained from $\Delta^k[n]$ by making the $(k-1)$st $(n-1)$-face and the $(k+1)$st $(n-1)$ face thin;
$\Delta^k[n]''$ be obtained from $\Delta^k[n]$ by making all $(n-1)$-faces thin.
An elementary anodyne extension in $Strat$, the category stratified simplicial sets is
or
for $n = 1,2, \cdots$ and $k \in [n]$.
A stratified simplicial set is a weak complicial set if it has the right lifting property with respect to all
$\Lambda^k[n] \stackrel{\subset_r}{\hookrightarrow} \Delta^k[n]$ and $\Lambda^k[n]' \stackrel{\subset_e}{\hookrightarrow} \Delta^k[n]''$
A complicial set is a weak complicial set in which such liftings are unique.
There is a model category structure that presents the (infinity,1)-category of weak complicial sets, hence that of weak $\omega$-categories. See
For $C$ a strict ω-category and $N(C)$ its ω-nerve, the Roberts stratification which regards each identity morphism as a thin cell makes $N(C)$ a strict complicial set, hence a weak complicial set. This example is not “saturated.”
There is also the stratification of $N(C)$ which regards each $\omega$-equivalence morphism as a thin cell. $N(C)$ with this stratification is a weak complicial set (example 17 of Ver06). This should be the “saturation” of the previous example, and exhibits the inclusion of strict $\omega$-categories into weak ones.
A simplicial set is a weak complicial set when equipped with its maximal stratification (every simplex of dimension $\gt 0$ is thin) if and only if it is a Kan complex. This example is, of course, saturated, and is viewed as embedding $\omega$-groupoids into $\omega$-categories.
A simplicial set is a quasi-category if and only if it is a weak complicial set when equipped with the stratification in which every simplex of dimension $\gt 1$ is thin, and only degenerate 1-simplices are thin. This example is not saturated; in its saturation the thin 1-simplices are the internal equivalences in a quasi-category (equivalently, those that become isomorphisms in its homotopy category). It presents the embedding of $(\infty,1)$-categories into weak $\omega$-categories.
Note that 1-simplex equivalences in a quasi-category are automatically preserved by simplicial maps between quasi-categories; this is why $QCat$ can “correctly” be regarded as a full subcategory of $sSet$. This is not true at higher levels; for instance not every simplicial map between nerves of strict $\omega$-categories necessarily preserves $\omega$-equivalence morphisms.
The definition of weak complicial sets is definition 14, page 9 of
Further developments are in